On an existence of a non-arithmetic diffused Borel measure in the real axis $R$. Let $V$ be a Polish vector space. A Borel measure $\mu$ in $V$ is called arithmetic measure if each measurable set $A$ with $\mu(A)>0$ has the following property: for an arbitrary $n >1$ there are $n$ points in $A$ which constitute an arithmetic progression. A Borel measure $\mu$ in $V$ is called diffused if it vanishes on singletons of $V$, i.e., $\mu(\\{x\\})=0$ for each $x \in V$. A linear Lebesgue measure in $R$ is an arithmetic measure. Notice that each arithmetic measure in the real axis $R$ is diffused. My question is: whether the converse is valid ? or Does there exist a $\sigma$-finite Borel diffused measure in $R$ which is not arithmetic?

There is a perfect set in R whose elements are Q-linearly independent - See here. Any (diffused) Borel measure supported on such a perfect set is a counterexample.

Notice that such a measure cannot be absolutely continuous w.r.t. Lebesgue measure.